
The functions \(a_1,\ldots , a_\ell \)  are tempered and have different growth rates, meaning \(a_1\prec a_2\prec \cdots \prec a_\ell \) , see [1]} (see also [2]} for a simpler argument that works for a slightly more restricted class of functions).

The functions \(\sum _{j=1}^\ell c_j a_j^{(k_j)}\)  are in \(\mathcal {T}+\mathcal {P}\)  for all \(c_1,\ldots , c_\ell \in {\mathbb {R}}\)  and \(k_1,\ldots , k_\ell \in {\mathbb {Z}}_+\) , see [3]}.

